The price to pay to cure the anomaly is that the edge index is defined along a loop of boundary conditions. This is actually sensible, since the bulk index does not pick out any distinguished boundary condition. Usually, and in particular in [1]}, a boundary condition is fixed when computing the number of edge modes, so there is no contradiction between the two statements (anomalous and anomaly-free). Here, the physical interpretation of the edge index is less obvious: it is a sort of boundary-driven pumping along the edge which is quantized after one cycle of boundary conditions. We are not aware of any physical evidence of such a quantity, but because of its stability there might be some way to observe it in some clever device.
